83. Roger Penrose, The Emperor’s New Mind (Oxford: Oxford University Press, 1989); also, Penrose, The Road to Reality (London: Cape, 2004).
84. In the language of quantum mechanics manuals, it is conventionally referred to as “measure.” Once again, there is something misleading about this language, inasmuch as it speaks about physics laboratories rather than speaking about the world.
85. The theorem of Tomita-Takesaki shows that a state on a von Neumann algebra defines a flow (a one-parameter family of modular automorphisms). Connes has shown that the flows defined by different states are equivalent up to internal automorphisms, and therefore define an abstract flow determined only by the noncommutative structure of algebra.
86. The internal automorphisms of the algebra referred to in the above note.
87. In a von Neumann algebra, the thermal time of a state is exactly the same as Tomita’s flow! The state is KMS with respect to this flow.
88. See Carlo Rovelli, “Statistical Mechanics of Gravity and the Thermodynamical Origin of Time,” Classical and Quantum Gravity 10 (1993): 1549–66; Alain Connes and Carlo Rovelli, “Von Neumann Algebra Automorphisms and Time-Thermodynamics Relation in General Covariant Quantum Theories,” Classical and Quantum Gravity 11 (1994): 2899–918.
89. Alain Connes, Danye Chéreau, and Jacques Dixmier, Le Théâtre quantique (Paris: Odile Jacob, 2013).
10. PERSPECTIVE
90. There are many confused aspects to this question. An excellent, cogent critique can be found in John Earman, “The ‘Past Hypothesis’: Not Even False,” Studies in History and Philosophy of Modern Physics 37 (2006): 399–430. In the text, “low initial entropy” is intended in the more general sense that, as Earman argues in this article, is far from being well understood.
91. Friedrich Nietzsche, The Gay Science (Cambridge, UK: Cambridge University Press, 2001), v–354.
92. The technical details can be found in Carlo Rovelli, “Is Time’s Arrow Perspectival?” in The Philosophy of Cosmology, eds. K. Chamcham et al. (Cambridge, UK: Cambridge University Press, 2017), https://arxiv.org/abs/1505.01125.
93. In the classical formulation of thermodynamics, we describe a system by specifying in the first place some variables on which we assume we can act from the outside (moving a piston, for example), or which we assume we can measure (a relative concentration of components, for xample). These are the “thermodynamic variables.” Thermodynamics is not a true description of the system; it is a description of these variables of the system—those through which we assume we are able to interact with the system.
94. For example, the entropy of the air in this room has a value taken from air as a homogeneous gas, but it changes (diminishes) if I measure its chemical composition.
95. A contemporary philosopher who has shed light on these aspects of the perspectival nature of the world is Jenann T. Ismael, The Situated Self (New York: Oxford University Press, 2007). Ismael has also written an excellent book on free wilclass="underline" How Physics Makes Us Free (New York: Oxford University Press, 2016).
96. David Z. Albert, in Time and Chance (Cambridge, MA: Harvard University Press, 2000), proposes to elevate this fact to a natural law, and calls it “past hypothesis.”
11. WHAT EMERGES FROM A PARTICULARITY
97. This is another common source of confusion, because a condensed cloud seems more “ordered” than a dispersed one. It isn’t, because the speed of the molecules of a dispersed cloud are all small (in an ordered manner), while, when the cloud condenses, the speeds of the molecules increase and spread in phase space. The molecules concentrate in physical space but disperse in phase space, which is the relevant one.
98. See, in particular, Stuart A. Kauffman, Humanity in a Creative Universe (New York: Oxford University Press, 2016).
99. The importance of the existence of this ramified structure of interactions in the universe for the understanding of the growth of local entropy is discussed, for instance, by Hans Reichenbach, in The Direction of Time (Berkeley: University of California Press, 1956). Reichenbach’s text is fundamental for whoever has doubts about these arguments or is interested in pursuing them in more depth.
100. On the precise relation between traces and entropy, see Reichenbach, The Direction of Time, in particular the discussion on the relation among entropy, traces, and common cause, and Albert, Time and Chance. A recent approach can be found in D. H. Wolpert, “Memory Systems, Computation and the Second Law of Thermodynamics,” International Journal of Theoretical Physics 31 (1992): 743–85.
101. On the difficult question of what “cause” means to us, see Nancy Cartwright, Hunting Causes and Using Them: Approaches in Philosophy and Economics (Cambridge, MA: Cambridge University Press, 2007).
102. “Common cause,” in Reichenbach’s terminology.
103. Bertrand Russell, “On the Notion of Cause,” Proceedings of the Aristotelian Society, N. S. 13 (1912–1913): 1–26.
104. Cartwright, Hunting Causes and Using Them.
105. For a lucid discussion on the question of the direction of time, see Huw Price, Time’s Arrow and Archimedes’ Point (Oxford: Oxford University Press, 1996).
12. THE SCENT OF THE MADELEINE
106. Milinda Pañha (The Questions of King Milinda) II.1, in T. W. Rhys Davids, Sacred Books of the East, vol. XXXV (Oxford: Clarendon Press, 1890).
107. Carlo Rovelli, Meaning = Information + Evolution, 2016, https://arxiv.org/abs/1611.02420.
108. G. Tononi, O. Sporns, and G. M. Edelman, “A Measure for Brain Complexity: Relating Functional Segregation and Integration in the Nervous System,” Proceedings of the National Academy of Sciences USA 91 (1994): 5033–37.
109. Jakob Hohwy, The Predictive Mind (Oxford: Oxford University Press, 2013).
110. See, for example, V. Mante, D. Sussillo, K. V. Shenoy, and W. T. Newsome, “Context-dependent Computation by Recurrent Dynamics in Prefrontal Cortex,” Nature 503 (2013): 78–84, and the literature cited in this article.
111. Dean Buonomano, Your Brain Is a Time Machine: The Neuroscience and Physics of Time (New York: Norton, 2017).
112. La Condemnation parisienne de 1277, ed. D. Piché (Paris: Vrin, 1999).
113. Edmund Husserl, Vorlesungen zur Phänomenologie des inneren Zeitbewusstseins (Halle: Niemeyer, 1928).
114. In the cited text, Husserl insists that this does not constitute a “physical phenomenon.” To a naturalist, this sounds like a statement of principle: he does not want to see memory as a physical phenomenon because he has decided to use phenomenological experience as the starting point of his analysis. The study of the dynamics of neurons in our brain shows how the phenomenon manifests itself in physical terms: the present of the physical state of my brain “retains” its past state, and this is gradually more faded the farther away we are from that past. See, for example, M. Jazayeri and M. N. Shadlen, “A Neural Mechanism for Sensing and Reproducing a Time Interval,” Current Biology 25 (2015): 2599–609.